Question

Rewrite the function in the form \(y=a(1+r)^t\) or \(y=a(1-r)^t\). Then state the growth or decay rate. \(y=a(2)^{t / 3}\)

Short answer

The rewritten function is \(y = a(\sqrt[3]{2})^t\), exhibiting growth at a rate of \(\sqrt[3]{2} - 1\).

Step-by-step solution

Identify the Base and Convert the Exponent

The given function is \(y = a(2)^{t/3}\), in which 2 is the base, and \(t/3\) is the exponent. The conversion to \(t\) form requires the equation \(2 = (1 + r)^3\). So, \(r = \sqrt[3]{2} - 1\). This is the compound growth rate.

Write the Function in the New Form

Now, substitute \(r\) into \(y = a(1 + r)^t\). This gives \(y = a(\sqrt[3]{2})^t\). This equation illustrates that \(y\) multiplies by \(\sqrt[3]{2}\) for each increment of \(t = 1\).

Determine the Growth Rate

As \(r = \sqrt[3]{2} - 1\) is positive, the function represents growth, not decay. The growth rate is therefore \(\sqrt[3]{2} - 1\).

Key concepts

Function Conversion

Converting functions is like translating one mathematical "language" into another. Here, we aim to rewrite the given function into a more recognizable form of exponential growth or decay. This makes it easier to identify key properties like the growth rate. The task is to take the function \( y = a(2)^{t/3} \) and express it in the form \( y = a(1 + r)^t \), which is more commonly used to portray exponential growth.In our function, "2" serves as the base, while "\(t/3\)" is an exponent. This base represents the factor by which the function grows or diminishes for each unit increment in time "t." The goal is to express this base through \(1 + r\) raised to an appropriate power. Solving the equation \( (1 + r)^3 = 2 \) helps us find the value of \( r \), which translates the base-2 growth to a percentage increase rate. From this conversion, \( r = \sqrt[3]{2} - 1 \), telling us that the new form will be \( y = a(\sqrt[3]{2})^t \). This shows how the original base is converted into the equivalent form involving the new growth multiplier \( \sqrt[3]{2} \).

Exponential Functions

Exponential functions model scenarios where quantities grow or shrink by a consistent percentage over equal increments of time. These are common in real-world settings like population growth, radioactive decay, and financial investments.For the function \( y = a(2)^{t/3} \), the growth behavior is determined by its base, which is "2". By converting it into the form \( y = a(1 + r)^t \), as we did earlier, we observe the growth factor '2' is expressed effectively as \( \sqrt[3]{2} \) per time unit. In practical terms:

• If the base is greater than 1, as in this case, it indicates growth. The quantity increases by a set rate for each time unit.
• If the base were between 0 and 1, it would imply decay — a decrease by a specific percentage for each time increment.

This knowledge aids in understanding how base numbers reflect the rate at which changes occur in exponential functions.

Growth Rate Calculation

Calculating the growth rate in an exponential function is essential for predicting how quickly values will rise or fall over time. Once you have the function in the form \( y = a(1 + r)^t \), pinpointing "r" becomes crucial. It represents the rate of growth (or decay) as a fraction.For \( y = a(2)^{t/3} \), we've translated it into \( y = a(\sqrt[3]{2})^t \) using the earlier conversion process, finding that \( r = \sqrt[3]{2} - 1 \).Knowing this:

• The growth rate is positive when \( r \) is greater than 0, indicating an increase.
• A negative \( r \) (though not applicable here) would signal decay.

In this example, \( \sqrt[3]{2} \approx 1.2599 \), so \( r = 0.2599 \) or approximately 25.99%. This number signifies that with each step in time, the function's value scales by about 25.99%, highlighting the dynamic nature of exponential growth. Understanding this helps in estimating how quickly quantities will escalate or decline in different scenarios.